In 2012 Gubeladze (Adv. Math. 2012)
introduced the notion of $k$-convex-normal polytopes to show
that
integral polytopes all of whose edges are longer than $4d(d+1)$
have
the integer decomposition property.
In the first part of this paper we show that for lattice polytopes
there is no difference between $k$- and $(k+1)$-convex-normality
(for
$k\geq 3 $) and improve the bound to $2d(d+1)$. In the second
part we
extend the definition to pairs of polytopes. Given two rational
polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement
of
the normal fan of $Q$.
If every edge $e_P$ of $P$ is at least $d$ times as long as the
corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap
\mathbb{Z}^d
= (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$.

This paper generalizes the Aleksandrov problem: the Mazur-Ulam
theorem on $n$-G-quasi normed spaces. It proves that a one-$n$-distance
preserving mapping is an $n$-isometry if and only if it has the
zero-$n$-G-quasi preserving property, and two kinds of $n$-isometries
on $n$-G-quasi normed space are equivalent; we generalize the
Benz theorem to n-normed spaces with no restrictions on the dimension
of spaces.

A commuting family of subnormal operators need
not have a commuting normal extension. We study when a representation
on an abelian semigroup can be extended to a normal representation,
and show that it suffices to extend the set of generators to
commuting normals. We also extend a result due to Athavale to
representations on abelian lattice ordered semigroups.

We use George Bergman's recent normal form for universally adjoining
an inner inverse to show that, for general rings, a nilpotent
regular element $x$ need not be unit-regular.
This contrasts sharply with the situation for nilpotent regular
elements in exchange rings (a large class of rings), and for
general rings when all powers of the nilpotent element $x$ are
regular.

An action of a Lie group $G$ on a smooth manifold $M$ is called
cohomogeneity one if the orbit space $M/G$ is of dimension $1$.
A Finsler metric $F$ on $M$ is called invariant if $F$ is
invariant under the action of $G$. In this paper,
we study invariant
Randers metrics on cohomogeneity one manifolds. We first give a
sufficient and necessary condition for the existence of invariant
Randers metrics on cohomogeneity one manifolds. Then we obtain
some results on invariant Killing vector fields on the
cohomogeneity one manifolds and use that to deduce some
sufficient and necessary condition for a cohomogeneity one
Randers metric to be Einstein.

Suppose that $G$ is a
finite group and $H$ is a subgroup of $G$. $H$ is said to be
$s$-semipermutable in $G$ if $HG_{p}=G_{p}H$ for any Sylow
$p$-subgroup $G_{p}$ of $G$ with $(p,|H|)=1$; $H$ is said to be
$s$-quasinormally embedded in $G$ if for each prime $p$ dividing the
order of $H$, a Sylow $p$-subgroup of $H$ is also a Sylow
$p$-subgroup of some $s$-quasinormal subgroup of $G$. We fix in
every non-cyclic Sylow subgroup $P$ of $G$ some subgroup $D$
satisfying $1\lt |D|\lt |P|$ and study the structure of $G$ under the
assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is either
$s$-semipermutable or $s$-quasinormally embedded in $G$.
Some recent results are generalized and unified.

We examine spectral behavior of irreducible tuples which do not
admit boundary property. In particular, we prove under some mild
assumption that the spectral radius of such an $m$-tuple $(T_1,
\dots, T_m)$ must be the operator norm of $T^*_1T_1 + \cdots +
T^*_mT_m$. We use this simple observation to ensure boundary
property for an irreducible, essentially normal joint $q$-isometry provided it
is not a joint isometry.
We further exhibit a family of
reproducing Hilbert $\mathbb{C}[z_1, \dots, z_m]$-modules (of which
the Drury-Arveson Hilbert module is a prototype) with the property that any
two nested unitarily equivalent submodules are indeed equal.

Let $g \geq 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\phi$ denote Euler's totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. We show that if $f$ is any function formed by composing $\phi$, $\sigma$, or $\lambda$, then the number
\[ 0. f(1) f(2) f(3) \dots \]
obtained by concatenating the base $g$ digits of successive $f$-values is $g$-normal. We also prove the same result if the inputs $1, 2, 3, \dots$ are replaced with the primes $2, 3, 5, \dots$. The proof is an adaptation of a method introduced by Copeland and ErdÅs in 1946 to prove the $10$-normality of $0.235711131719\ldots$.

Let $g \geq 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\phi$ denote Euler's totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. We show that if $f$ is any function formed by composing $\phi$, $\sigma$, or $\lambda$, then the number
\[ 0. f(1) f(2) f(3) \dots \]
obtained by concatenating the base $g$ digits of successive $f$-values is $g$-normal. We also prove the same result if the inputs $1, 2, 3, \dots$ are replaced with the primes $2, 3, 5, \dots$. The proof is an adaptation of a method introduced by Copeland and ErdÅs in 1946 to prove the $10$-normality of $0.235711131719\ldots$.

We establish that if it is consistent that there is a
supercompact cardinal, then it is consistent that every locally
compact, hereditarily normal space which does not include a perfect
pre-image of $\omega_1$ is hereditarily paracompact.

In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$.

To every subnormal $m$-variable weighted shift $S$ (with bounded
positive weights) corresponds a positive Reinhardt measure $\mu$
supported on a compact Reinhardt subset of $\mathbb C^m$. We show that, for
$m \geq 2$, the dimensions of the $1$-st cohomology vector spaces
associated with the Koszul complexes of $S$ and its dual ${\tilde S}$
are different if a certain radial function happens to be integrable
with respect to $\mu$ (which is indeed the case with many classical
examples). In particular, $S$ cannot in that case be similar to
${\tilde S}$. We next prove that, for $m \geq 2$, a Fredholm subnormal
$m$-variable weighted shift $S$ cannot be similar to its dual.

In this paper we prove weighted norm inequalities with weights in
the $A_p$ classes, for pseudodifferential operators with symbols in
the class ${S^{n(\rho -1)}_{\rho, \delta}}$ that fall outside the
scope of CalderÃ³n-Zygmund theory. This is accomplished by
controlling the sharp function of the pseudodifferential operator by
Hardy-Littlewood type maximal functions. Our weighted norm
inequalities also yield $L^{p}$ boundedness of commutators of
functions of bounded mean oscillation with a wide class of operators
in $\mathrm{OP}S^{m}_{\rho, \delta}$.

We give precise conditions under which the composition
of a norm with a convex function yields a
uniformly convex function on a Banach space.
Various applications are given to functions of power type.
The results are dualized to study uniform smoothness
and several examples are provided.

We provide a criterion for the central norm to be
any value in the simple continued fraction expansion of $\sqrt{D}$
for any non-square integer $D>1$. We also provide a simple criterion
for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of
congruence conditions modulo $D$.

We determine the best constants $C_{p,\infty}$ and $C_{1,p}$,
$1 < p < \infty$, for which the following holds. If $u$, $v$ are
orthogonal harmonic functions on a Euclidean domain such that $v$ is
differentially subordinate to $u$, then
$$ \|v\|_p \leq C_{p,\infty}
\|u\|_\infty,\quad
\|v\|_1 \leq C_{1,p} \|u\|_p.
$$
In particular, the inequalities are still sharp for the conjugate
harmonic functions on the unit disc of $\mathbb R^2$.
Sharp probabilistic versions of these estimates are also studied.
As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.

We extend the notion of Zindler curve from the Euclidean plane to
normed planes. A characterization of Zindler curves for general
normed planes is given, and the relation between Zindler curves and
curves of constant area-halving distances in such planes is
discussed.

For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely
bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We
characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm
one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we
describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize
those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly
relax the norm bounds.)

We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.

We generalize LÃ¶wner's method for proving that matrix monotone
functions are operator monotone. The relation $x\leq y$ on bounded
operators is our model for a definition of $C^{*}$-relations
being residually finite dimensional.
Our main result is a meta-theorem about theorems involving relations
on bounded operators. If we can show there are residually finite dimensional
relations involved and verify a technical condition, then such a
theorem will follow from its restriction to matrices.
Applications are shown regarding norms of exponentials, the norms
of commutators, and "positive" noncommutative $*$-polynomials.

We show a pointwise estimate for the Fourier
transform on the line involving the number of times the function
changes monotonicity. The contrapositive of the theorem may be used to
find a lower bound to the number of local maxima of a function. We
also show two applications of the theorem. The first is the two weight
problem for the Fourier transform, and the second is estimating the
number of roots of the derivative of a function.

This paper deals with the analytic solvability of a special class of
complex vector fields defined on the real plane, where they are
tangent to
a closed real curve, while off the real curve, they are elliptic.